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Transcript
ICS 278: Data Mining
Lecture 5: Regression Algorithms
Padhraic Smyth
Department of Information and Computer Science
University of California, Irvine
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Notation
• Variables X, Y….. with values x, y (lower case)
• Vectors indicated by X
• Components of X indicated by Xj with values xj
• “Matrix” data set D with n rows and p columns
– jth column contains values for variable Xj
– ith row contains a vector of measurements on object i, indicated by x(i)
– The jth measurement value for the ith object is xj(i)
• Unknown parameter for a model = q
– Can also use other Greek letters, like a, b, d, g ew
– Vector of parameters = q
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Example: Multivariate Linear Regression
• Task: predict real-valued Y, given real-valued vector X
• Score function, e.g.,
S(q) =
Si [y(i) – f(x(i) ; q) ]2
• Model structure: f(x ; q) = a0 +
S aj xj
• Model parameters = q = {a0, a1, …… ap }
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
S = S e2 = e’ e
= (y – X a)’ (y – X a)
= y’ y – a’ X’ y – y’ X a + a’ X’ X a
= y’ y – 2 a’ X’ y + a’ X’ X a
Taking derivative of S with respect to the components of a gives….
dS/da = -2X’y + 2 X’ X a
Set this to 0 to find the extremum (minimum) of S as a function of a…
 - 2X’y + 2 X’ X a = 0
 X’Xa = X’ y
Letting X’X = C, and X’y = b, we have C a = b, i.e., a set of linear equations
We could solve this directly by matrix inversion, i.e.,
a = C-1 b = ( X’ X )-1 X’ y
…. but there are better ways to do this
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Comments on Multivariate Linear Regression
• prediction is a linear function of the parameters
• Score function: quadratic in predictions and parameters
 Derivative of score is linear in the parameters
 Leads to a linear algebra optimization problem, i.e., Ca = b
• Model structure is simple….
– p-1 dimensional hyperplane in p-dimensions
– Linear weights => interpretability
• Useful as a baseline model
– to compare more complex models to
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Limitations of Linear Regression
• True relationship of X and Y might be non-linear
– Suggests generalizations to non-linear models
• Complexity:
– O(p3) - could be a problem for large p
• Correlation among the X variables
– Can cause numerical instability (C may be ill-conditioned)
– Problems in interpretability (identifiability)
• Includes all variables in the model…
– What if p=100 but only 3 variables are related to Y?
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Finding the k best variables
• Find the subset of k variables that predicts best:
– This is a generic problem when p is large
(arises with all types of models, not just linear regression)
• Now we have models with different complexity..
–
–
–
–
E.g., p models with a single variable
p(p-1)/2 models with 2 variables, etc…
2p possible models in total
Note that when we add or delete a variable, the optimal weights on the
other variables will change in general
• k best is not the same as the best k individual variables
• What does “best” mean here?
– Return to this later
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Search Problem
• How can we search over all 2p possible models?
– exhaustive search is clearly infeasible
– Heuristic search is used to search over model space:
• Forward search (greedy)
• Backward search (greedy)
• Generalizations (add or delete)
– Think of operators in search space
• Branch and bound techniques
– This type of variable selection problem is common to many data
mining algorithms
• Outer loop that searches over variable combinations
• Inner loop that evaluates each combination
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Defining what “best” means
• How do we measure “best”?
– Best performance on the training data?
• K = p will be best (i.e., use all variables)
• So this is not useful
• Note:
– Performance on the training data will in general be optimistic
• Alternatives:
– Measure performance on a single validation set
– Measure performance using multiple validation sets
• Cross-validation
– Add a penalty term to the score function that “corrects” for optimism
• E.g., GCV
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Empirical Learning
• Squared Error score (as an example: we could use other scores)
S(q) =
Si [y(i) – f(x(i) ; q) ]2
where S(q) is defined on the training data D
• We are really interested in finding the f(x; q) that best predicts y on
future data, i.e., minimizing
E [S] = E [y – f(x ; q) ]2
• Empirical learning
– Minimize S(q) on the training data Dtrain
– If Dtrain is large and model is simple we are assuming that the best f
on training data is also the best predictor f on future test data Dtest
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Complexity and Generalization
Score
Function
Stest(q)
Strain(q)
Optimal model
complexity
Data Mining Lectures
Lecture 5: Regression
Complexity = degrees
of freedom in the model
(e.g., number of variables)
Padhraic Smyth, UC Irvine
Using Validation Data
Training Data
Validation Data
Test Data
Data Mining Lectures
Use this data to find the best q
for each model fk(x ; q)
Use this data to
(1) calculate an estimate of Sk(q) for
each fk(x ; q) and
(2) select k* = arg mink Sk(q)
Use this data to calculate an
unbiased estimate of Sk*(q) for
the selected model
Lecture 5: Regression
Padhraic Smyth, UC Irvine
2 different (but related) issues here
• 1. Finding the function f that minimizes S(q) for future
data
• 2. Getting a good estimate of S(q), using the chosen
function, on future data,
– e.g., we might have selected the best function f, but our
estimate of its performance will be optimistically biased if our
estimate of the score uses any of the same data used to fit and
select the model.
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Non-linear models, linear in parameters
•
We can add additional polynomial terms in our equations, e.g., all “2nd
order” terms
f(x ; q) = a0 +
•
S aj xj + S bij xi xj
Note that it is a non-linear functional form, but it is linear in the parameters
(so still referred to as “linear regression”)
– We can just treat the xi xj terms as additional fixed inputs
– In fact we can add in any non-linear input functions!, e.g.
f(x ; q) = a0 +
S aj fj(x)
Comments:
- Exact same linear algebra for optimization (same math)
- size of model space has now exploded -> very large search problem
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Non-linear (both model and parameters)
•
We can generalize further to models that are nonlinear in all aspects
f(x ; q) = a0 +
S ak gk(bk0 +S bkj xj )
where the g’s are non-linear functions with fixed functional forms.
In machine learning this is called a neural network
In statistics this might be referred to as a generalized linear model or
projection-pursuit regression
For almost any score function of interest, e.g., squared error, the score
function is a non-linear function of the parameters.
Closed form (analytical) solutions are rare.
Thus, we have a multivariate non-linear optimization problem
(which may be quite difficult!)
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Optimization of a non-linear score function
• We seek the minimum of a function in d dimensions, where d is the
number of parameters (d could be large!)
• There are a multitude of heuristic search techniques (see chapter 8)
–
–
–
–
–
–
Steepest descent (follow the gradient)
Newton methods (use 2nd derivative information)
Conjugate gradient
Line search
Stochastic search
Genetic algorithms
• Two cases:
– Convex (nice -> means a single global optimum)
– Non-convex (multiple local optima => need multiple starts)
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Other non-linear models
• Splines
– “patch” together low-order polynomials
– Works well in 1 dimension, less well in higher dimensions
• Memory-based models
y’ =
S w(x’,x) y,
where y’s are from the training data
w(x’,x) = function of distance of x from x’
• Local linear regression
y’ = a0 +
S aj xj
, where the alpha’s are fit at prediction
time just to the (y,x) pairs that are close to x’
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Time-series prediction as regression
• Measurements over time x1,…… xt
• We want to predict xt+1 given x1,…… xt
• Autoregressive model
f( x1,…… xt ; q ) =
S ak xt-k
– Number of coefficients K = memory of the model
– Can take advantage of regression techniques in general to solve this
problem (e.g., linear in parameters, score function = squared error, etc)
• Generalizations
– Vector x
– Non-linear function instead of linear
– Add in terms for time-trend (linear, seasonal), for “jumps”, etc
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Other aspects of regression
• Diagnostics
– Useful in low dimensions
• Weighted regression
– Useful when rows have different weights
• Different score functions
– E.g. absolute error
• Predicting y values constrained to a certain range
• Predicting binary y values
– Regression as a generalization of classification
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Classification as a special case of Regression
• Let y take values 0 or 1
• Regression (in general) tries to learn an f function that matches
E[y|x]
• For binary y we have
E[y|x] = 1. p(y=1|x) + 0 p(y=0|x) = p(y=1|x)
• Thus, if regression drives f towards E[y|x], then for binary
classification problems a regression model will try to approximate
p(y=1|x) (posterior class probabilities)
– e.g., this is what neural networks do
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Predicting an output between 0 and 1
•
We often have a problem where y lies between 0 and 1
– probability that a person with attributes X will survive 10 years
– proportion of people in Zip code X who will buy a product
•
We could use linear regression, but…..
•
Instead we can use the logistic function
log p(y=1|x)/log p(y=0|x) = a0 +
Equivalently,
p(y=1|x) = 1/[1 + exp(- a0 -
S aj xj
S aj xj
)]
We model the log-odds as a linear function of the input variables. This is
known as logistic regression.
Could be viewed as a simple neural network with a single hidden unit
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Logistic Regression
•
How do we estimate the alpha parameters?
– Problem: linear algebra solution no longer applies since our function form
(and score) is now non-linear in the parameters
•
Common approach
– Score function = likelihood = probability of observed data
– Select parameters to maximize the log-likelihood (“maximum likelihood”)
– S(q) =
Si log p( y(i) | x(i) ; a)
= Si y(i) log p( y(i)=1| x(i) ; a) + [1-y(i)] log(1- p( y(i)=1| x(i) ; a))
– Can compute the 2nd derivative directly as weighted matrix
• Forms the basis for an iterative 2nd order Newton scheme
• Known as iteratively reweighted least-squares
• Log-likelihood here is convex: so it is quite stable (only one global maximum!).
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Tree-Structured Regression
• Functional form of model is a “regression tree”
–
–
–
–
Univariate thresholds at internal nodes
Constant or linear surfaces at the leaf nodes
Yields piecewise constant (or linear) surface
(like classification tree, but for regression)
• Very crude functional form…. But
– Can be very useful in high-dimensional problems
– Can be very interpretable
• Search problem
– Finding the optimal tree is NP-hard (for any reasonable score)
– Practice: greedy algorithms
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
More on regression trees
• Greedy search algorithm
– For each variable, find the best split point such the mean of Y
either side of the split minimizes the mean-squared error
– Select the variable with the minimum average error
• Partition the data using the threshold
– Recursively apply this selection procedure to each “branch”
• What size tree?
– A full tree will likely overfit the data
– Common methods for tree selection
• Grow a large tree and select an appropriate subtree by Xvalidation
• Grow a number of small fixed-sized trees and average their
predictions
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Model Averaging
• Can average over parameters and models
– Why? Any point estimate of parameters or a single model has only a
small chance of the being the best
– Averaging makes our predictions more stable and less sensitive to
random variations in a particular data set (good for less stable models
like trees)
• Model averaging flavors
– Fully Bayesian: parameters and models
– “empirical Bayesian”: learn weights over multiple models
• E.g., stacking and bagging
– Build multiple simple models in a systematic way and combine them,
e.g., boosting
– Will say more about this in later lectures
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Components of Data Mining Algorithms
• Representation:
– Determining the nature and structure of the representation to be
used;
• Score function
– quantifying and comparing how well different representations fit
the data
• Search/Optimization method
– Choosing an algorithmic process to optimize the score function;
and
• Data Management
– Deciding what principles of data management are required to
implement the algorithms efficiently.
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
What’s in a Data Mining Algorithm?
Task
Representation
Score Function
Search/Optimization
Data
Management
Models,
Parameters
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Multivariate Linear Regression
Task
Representation
Data Mining Lectures
Regression
Y = Weighted linear sum
of X’s
Score Function
Least-squares
Search/Optimization
Linear algebra
Data
Management
None specified
Models,
Parameters
Regression
coefficients
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Autoregressive Time Series Models
Task
Representation
Data Mining Lectures
Time Series Regression
X = Weighted linear sum
of earlier X’s
Score Function
Least-squares
Search/Optimization
Linear algebra
Data
Management
None specified
Models,
Parameters
Regression
coefficients
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Neural Networks
Task
Representation
Regression
Y = nonlin function of X’s
Score Function
Least-squares
Search/Optimization
Data Mining Lectures
Gradient descent
Data
Management
None specified
Models,
Parameters
Network
weights
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Logistic Regression
Task
Representation
Regression
Log-odds(Y) = linear
function of X’s
Score Function
Search/Optimization
Data Mining Lectures
Log-likelihood
Iterative (Newton) method
Data
Management
None specified
Models,
Parameters
Logistic
weights
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Software
• MATLAB
– Many free “toolboxes” on the Web for regression and prediction
– e.g., see http://lib.stat.cmu.edu/matlab/
and in particular the CompStats toolbox
• R
– General purpose statistical computing environment (successor to S)
– Free (!)
– Widely used by statisticians, has a huge library of functions and
visualization tools
• Commercial tools
– SAS, other statistical packages
– Data mining packages
– Often are not progammable: offer a fixed menu of items
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Suggested Reading in Text
•
Chapter 4:
•
Chapter 5:
•
Chapter 6:
•
Chapter 8:
•
Chapter 9:
– General statistical aspects of model fitting
– Pages 93 to 116, plus Section 4.7 on sampling
– “reductionist” view of learning algorithms (can skim this)
– Different forms of functional forms for modeling
– Pages 165 to 183
– Section 8.3 on multivariate optimization
– linear regression and related methods
– Can skip Section 11.3
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine
Good References
N. R. Draper and H. Smith, Applied Regression Analysis, 2nd edition,
Wiley, 1981
(the “bible” for classical regression methods)
T. Hastie, R. Tibshirani, and J. Friedman, Elements of Statistical Learning,
Springer Verlag, 2001
(statistically-oriented overview of modern ideas in regression and
classification)
Data Mining Lectures
Lecture 5: Regression
Padhraic Smyth, UC Irvine